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We perform a case study of the behavior of gluon radiation beyond the softapproximation, using as an example the Drell-Yan production cross section atNNLO. We draw a careful distinction between the eikonal expansion, which is inpowers of the soft gluon energies, and the expansion in powers of the thresholdvariable $1 - z$, which involves important hard-collinear effects. Focusing onthe contribution to the NNLO Drell-Yan K-factor arising from real-virtualinterference, we use the method of regions to classify all relevantcontributions up to next-to-leading power in the threshold expansion. With thismethod, we reproduce the exact two-loop result to the required accuracy,including $z$-independent non-logarithmic contributions, and we preciselyidentify the origin of the soft-collinear interference which breaks simplesoft-gluon factorization at next-to-eikonal level. Our results pave the way forthe development of a general factorisation formula for next-to-leading-powerthreshold logarithms, and clarify the nature of loop corrections to a set ofrecently proposed next-to-soft theorems.

The method of regions and next-to-soft corrections in Drell-Yan production